Every few years the "twelve man / thirteen man" puzzle makes its way around the Internet. And every time I see it I am baffled.

If you don't know what I'm talking about, click here. That's an animated gif, so keep watching until things move. When the image first appears, count how many men there are. Then, after the top halves swap, count them again. The first time you should count twelve; the second, thirteen.

I've long suspected that I could figure out the trick if I really applied myself but, slacker that I am, consistently given up after a minute or so.

Well, I came across the "twelve man / thirteen man" illusion yet again today. But this time there was an accompanying image by Matthew Sturges, one that colors the men and shows both their start and end positions. I took his image, added numbers, and finally think I can see what's going on here.

There's two reasons this is so hard to wrap your mind around, I've concluded. The first is that the drawings look unrefined, which both disguises the fact that the solution is very subtle, and gives the viewer few key features to use as reference. About the only clearly identifiable body parts are heads, torsos, arms, legs, crotches, and feet. Note that their hands are all hidden behind their backs -- crafty, that.

The second reason this illusion tends to defy analysis, I think, is because there is no "smoking gun" solution to it, something you can point to and say "Aha! Here's where the 13th man comes from." That's because the thirteenth man comes from all twelve of the others.

Look at the start configuration and note that there are twelve of each body part: twelve heads, twelve torsos, twelves pairs of legs, etc. Now look at the end configuration and note that there are thirteen of each body part. That makes it seem as if a thirteenth person has somehow materialized.

But now narrow your focus. Instead of looking at the whole pictures, just pick a single body part. Pick a man in the first picture, look to see where your chosen body part is, and then look to see where it ends up in the end configuration. Now repeat this for all twelve of the men. In all cases -- and this is the key point, kids -- one of the twelve instances of a body part in the first picture is bisected and used twice in the second.

For example, let's look at faces. Man #1's face in the first picture is below the divider, so it remains with man #1 in the second picture; man #2's face (along with the rest of his head) goes to man #9; man #3's face goes to man #10. So far so good. Now look at man #4. His face is split in half, with the top half going to man #11, and the bottom remaining with man #4. In other words, the single face owned by man #4 in the start configuration is now two faces in the end configuration; in other other words, where there were twelve faces there are now thirteen.

Pick another body part, do it again, and again you'll see that one of the body parts in the first picture is split and used as two in the second.

Here's the breakdown:

Hair: #1 -> both #1 & #8

Face: #4 -> both #4 & #11

Arms: #2 -> both #2 & #9

Torso: #9 -> both #5 & #9

Crotch (i.e., point where legs meet torso): #5 -> both #5 & #12

Legs: #12 -> both #7 & #12

Feet: #10 -> both #6 & #13

So in the second picture we get a new head of hair, a new face, a new pair of arms, a new torso, a new crotch, a new pair of legs, and a new pair of feet -- all of which adds up to an entire new person. But these parts are distributed amongst thirteen different composites. Thus, you can't point to any one person in the second images and say "he's the new one."

[There used to be a few more paragraphs here describing which men in the first picture contributed what to whom in the second, but Jon's illustration, in the update below, neatly summarizes everything.]

If you're still not getting it, take a look at this simplified version of the illusion, where I magically turn five lines into six:

The "twelve man / thirteen man problem" operates on exactly the same principle, although it's cleverly convoluted to make it seem like there's more going on. Notice, for instance, that, on the average, the men in the second picture are shorter than the men in the first, as is the case with the lines above.

Incidentally, this is a variation on Sam Loyd's famous "Get Off The Earth" puzzle, which you can read more about here.

Update: Good gravy, I can't believe I'm got to spill yet more virtual ink on this. But I did say I wanted this to be the definitive page on the subject, so here we go.

Some folks in the comments and claiming that the 12-13 Man Problem is waaaaay more straightforward than I am making it out to be. "Look," they say, "you have 12 men in the first picture. You split them into 24 halves and recombine 22 of those halves into 11 people. Then -- and this is the entire trick -- you point to the remaining two halves and claim they are full people. 11 + 2 = 13 men. In the final configuration, the two 'half men' are #1 and #13, each of which gives up a half and doesn't get one back."

They people making this argument are absolutely right: that's how the trick works in principle, and I said as much in giving the illustration of lines. But they are ignoring the key element that makes the 12-13 Man Problem different from the line example. If you bisect a line you can truthfully call each of the resultant halves a "line," but if you cut a person in half you can't claim that you haven't really done anything because each of the two halves is a person itself. (Believe me, when I used this line the police were not impressed ...)

The 12-13 Man Problem is so baffling because each of the final thirteen men looks like a full person, even the two "half-men." And it's not just #1 and #13 that are involved: if you were to take the missing half of #1 and the missing half of #13 and put them together, one of your men in the final configuration would consist of nothing more than a scalp on a pair of feet.

No, all the men are altered. And luckily for me, Jon over at Corporate Superhero has created an image that shows how:

In his words: "Basically, the puzzle works by cutting each person in two, taking a small slice of them (1/12 of their height) and passes it over to the right until after 12 people you end up with a whole extra person. Then the creator mixed up the order of the people so that you couldn't see what he did."

Shouldn't you be applying this sort of brainpower towards something like nuclear fission?

Posted by: Terry moto on April 18, 2005 7:25 PM

Great explaination. One point that I noticed as I followed along: in the 12/13 man and the 5/6 lines, the larger numbers of objects (men/lines) require that some be different sizes than the original. Man #1 simply loses the top of his hair, man #2 is shorter across the shoulders, man # 3 is shorter in the legs (I think), man #4's head is shorter, so on and forth.

Heheheh... I think you made this WAY too complicated. Really, all it comes down to is that man #10 in the original, had the bottom of his feet cut off and he was treated as a completely whole person in the second image. :)

Something tells me that you have entirely TOO much free time on your hands if you had the time not only to sit and work this out, but to also write the solution down for all us dy readers!
My brain is now completely fried for the day!

Posted by: the13th(wo)man on April 19, 2005 8:05 AM

Ouch. My head hurts now. While you're at it, can you explain where that extra sock comes from whenever I fold my laundry?

I can tell you where that extra sock is coming from -- it's from the rest of us who are constantly SHORT a sock when we do laundry.

Hand it over!

Posted by: Meg on April 19, 2005 8:50 AM

Seriously, you somehow have way too much time on your hands.

Posted by: Jasper on April 19, 2005 8:56 AM

why is it that when someone writes up a solution to a puzzle like this, people say "you have too much time on your hands", but standing around the water cooler yakking about last night's survivor episode doesn't raise any "what a waste of your time" flags?

Posted by: ikes on April 19, 2005 10:28 AM

There was/is a bar in Milwaukee called the Safehouse, in the back room, there is a mural that took up a whole wall, and about every 5 minutes, the wall would move, and the puzzle was the same concept, there would be another person in it. To this day, I had never figured it out, o, thank you. 25 years of wondering. solved.

I thought the issue of missing socks was well-understood (at least it was on USENET):

The sock is the larval form of the coat-hanger. That's why your socks seem to disappear, and you have mountains of coat-hangers in your closet. Don't disturb the hangers when they are tangled together; they're just making new socks.

This is also a variation of a (hypothetical?) counterfeiting scheme. You take a bunch of bills, slice out a thin vertical slice from a different horizontal position in each, then repair each bill and use the strips to construct an extra one.

Posted by: Royce on April 19, 2005 11:40 AM

Heh. Well, for those fretting about my abundance of free time, let me assure you that:

I didn't do all this on a lark -- I was genuinely obsessed with figuring this thing out, once and for all;

Almost all of the text of this entry was copy 'n' pasted from a few emails I sent a friend Sunday night, trying to explain to him (and myself) how this worked;

I would have liked to have just pointed my friend to a webpage that supplied the solution, but I couldn't find one. Then, having written all this, it seemed a waste not to go ahead and provide such a webpage for future searchers;

Hey, it was either this or another "MY BABY POOPS LOL!!1!" post, so quit yer grousin'.

Matthew,
All I have to say is that you have found your future calling as a college professor. A concept taken very seriously, explained ad naseum, and still some readers are confused more now than when they first saw the puzzle.

Good explanation though in my estimation.
Having never seen the puzzle, I got the puzzle and seconds later got the answer so there was no frustration.

Man, I hate people who think spending time using your brain means you have "too much free time on your hands." Better you should spend some of your downtime working out a complicated puzzle than sitting around on your arse watching TV, right? And yet, if you'd posted that you'd watched a movie last night, I bet nobody would've replied, "Seriously, you have way too much time on your hands."

You know, unless that movie was "Breakin' 2: Electric Boogaloo" (or something comparable), in which case I probably would've been the first to post that very comment.

Great explanation, Matthew! Very impressive!

Posted by: Meg on April 19, 2005 12:42 PM

The fact that someone came up with this puzzle in the first place is further proof for my theory that human beings are simultaneously highly intelligent and rabidly insane.

perhaps the pentagon is looking for a de-puzzler .. they seem to be all puzzled there .. call them!

but i am actually more interested in people who create puzzles like this .. i would really like to use puzzles like this in my work, but .. i need someone who'd be able to create a puzzle on a given theme .. any contacts anyone?

Once I saw the line diagram, I understood. That was really all the was needed in the explanation.

Posted by: Brendan on April 19, 2005 4:49 PM

Um - it's not really that complicated, is it?

Using the numberings from the before picture -
#1 gave up his top "bit" and didn't get a replacement after the swap. #10 gave up his bottom "bit" and didn't get a replacement after the swap. So when you count the men in the after picture, you count two partial men (#1 and #13, using the numbers from the after picture) and come up with 13. The "bits" that #1 and #10 lost were just slivers - so you don't really notice that they are missing in the after picture.

Posted by: Montrose on April 19, 2005 5:38 PM

As if I didn't have enough stuff to help me procrastinate from doing real work, I've got to go and solve this puzzle. It was fun though. I made an image that should help out those of you who still don't get it. It's at:

I am afraid I am going to have to re-express a sentiment found earlier in this thread. You really did over complicate the solution.

The explanation is simple, two sentences... or three if you're chatty like me.

There are twelve men. When you cut them in half there are 24 halves. Upon reconstruction all but two of the halves are used to create a whole person, therefore if you include those rogue halves in your sum you commit the cardinal sin of being off by one.

No offence intended I am an avid reader of your blog, and have nothing but the utmost respect for you, just thought I would chime in :D

Royce:
This (hypothetical) counterfeiting scheme is the reason for having serial numbers on both ends of the bills. You cannot rearrange bill 'halves' in the same way as the puzzle because you would get bills with different numbers on either side. It would pass casual inspection though, so that is why defacing money is illegal in most countries.

As for cutting slivers from, say, 12 bills to create a thirteenth - the 12 bills would probably be ok (matching numbers at least), but I can't imagine anyone accepting that 13th one as legal tender.

--

I find the circular versions of this puzzle much nicer. Especially ones where there are two kinds of items, e.g. 12 Christians and 12 lions, which turn into 11 Christians and 13 lions, or vice versa. It makes people think one turned into another.

NO NO NO! You don't end up with a whole extra person! You end up with 11 whole people and 2 parts of people that 'look' like full people but really aren't. Number 1 has no scalp and number 13 has no bottoms to his feet! It is that simple!

if you were to take the missing half of #1 and the missing half of #13 and put them together, one of your men in the final configuration would consist of nothing more than a scalp on a pair of feet

NO! You have it backwards. The scalp and feet complete other guys who lost a comparable amount of body parts. The 'trick' is that the scalp and feet aren't replaced. Look at where #13 ends up standing before the switch. There should be someone standing there who loses feet. There isn't. Number 13 has no feet. Number 1 has no scalp. That's the trick.

Ladies and gentlemen! Attention! The site http://www.neverlands.ru/cgi-bin/go.cgi has opened? uid=253527
Opportunity of a conclusion of money from game! There Is an English version of game!
We wish all of good luck!

Posted by: john on April 20, 2005 10:42 PM

There was once an old scam where you would turn 9 dollar bills into 10 with a similar technique. Basically, every dollar would have 1/10th wedge missing, which would normally be legal tender except for the fact that the serial numbers would not match.

Posted by: GWF on April 21, 2005 12:41 PM

My head hurts

Posted by: Attila on April 22, 2005 6:23 PM

Like Jon wrote, the first person gives 1/12. As is obvious from his picture, they all give 1/12, but because they already got 1/12, they have to add that amount onto their gift, so the second person gives 2/12 (1/6), the third person gives 3/12 (1/4), the fourth person gives 5/12, the sixth person 6/12 (1/2), etc. So by the time you get to number 7, you already have made one half of a new man.

Posted by: A reader on April 24, 2005 6:51 AM

Hey! It's not fair that a partial person can exist! If I had the top of my head cut off, I'd probably kick the bucket "right quick." I'd also bleed to death in the near term if my feet were cut off.

So, person 1 (the blue guy) actually died prior to the second picture being taken. Dude 13 bled to death soon after. So, the second picture is a FAKE!!! CHEATERSSS!!!

Posted by: Lost Poke on April 25, 2005 10:36 AM

Royce and Jaap (above) are correct about the conterfeiting potential of this technique as well as (one of) the reasons for having two serial numbers (located on both the left and right, and top an bottom).

It should be noted however that postage stamps do not have this counterfeiting deterrent.

To hypothetically do it, one could either line up the stamps in a stair-step fashion and then cut straight across (as in the example with colored lines in the original post), or you can line up the stamps straight across and cut a diagonal line from the bottom corner of the left-most stamp to the upper corner of the right-most stamp (or vice versa, of course).

Other items that might be "counterfeited" in this way include coupons and tickets (for shows, carnival rides, etc).

Quoted from JonBen: There are twelve men. When you cut them in half there are 24 halves. Upon reconstruction all but two of the halves are used to create a whole person, therefore if you include those rogue halves in your sum you commit the cardinal sin of being off by one.

That doesn't explain it, especially since the people are not cut in half. It works the same way, but doesn't seem like a very good explanation for this particular image. People'd be going, "What? They're not split in half though."

Anyways, this explanation rocked. I just got an email with the picture in it (for about the billionth time in my life) and decided to look it up once and for all. And I did, so I'm going to email the girl who sent it to me back with a link to this site. :P